159-161] CIRCULAR VORTICES. 255 



Let us introduce a symbol x , defined by 



It is plain that the position of the circle (# , tn- ) will depend only 

 on the strengths and the configuration of the vortices, and not on 

 the position of the origin on the axis of symmetry. This circle 

 may be called the circular axis of the whole system of vortex 

 rings ; we have seen that it remains constant in radius. To find 

 its motion parallel to Ox, we have from (6) and (4), 



(7), 



since u and v are the rates of increase of x and w for any 

 particular vortex. By means of (5) we can put this in the form 



] T 



(x-x.)vn&amp;gt; ............ (8), 



which will be of use to us later. The added term vanishes, since 

 = on account of the constancy of the mean radius. 



161. On account of the symmetry about Ox, there exists, in 

 the cases at present under consideration, a stream-function ty, 

 defined as in Art. 93, viz. we have 



1 dilr 

 ~^r 



TV dx 



, dv du 1 /cfiilr d 2 ^ 1 d\lr\ 



whence 2o&amp;gt; = -= = r. + .11 n (2). 



dx dm ty \ dx* d^ -BT dwj 



It appears from Art. 148 (4) that at a great distance from the 

 vortices u, v are of the order R~ s , and therefore ty will be of the 

 order R~ l . 



The formula for the kinetic energy may therefore be written 

 T = 7rpff(u* + ir) or dxdvr 



f/y ^^ ^ */ r ^ 7 ^ 



= TTO In v j- u ~- dxaiz 

 Jj\ dx dfrj 



by a partial integration, the terms at the limits vanishing. 



